During the Napoleonic Wars, Some troops performed as "mounted infantry", i.e. they rode horses to move, but dismounted and fought like infantry. Dragoons are usually given as examples. How was this force actually used in battle? Cavalry normally use their speed and momentum as advantage, e.g. by charging into infantry flank or rear. But dragoons kind of gave away this advantage by dismounting and fighting like infantry.

Also since they are train to fight like infantry, they most probably will be easily defeated by regular cavalry. And since infantry is normally more numerous and fire better than cavalry, they would be outgunned by regular infantry. In any case, there were no advantage.

How was it used in battle? Any example of battles in this era where mounted infantry played an important part?

The answer may differ depending on the country. Mike Duncan's Revolutions has a very nice discussion of how Dragoons were employed in the English Civil War. They resisted dismounting.
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Mark C. WallaceNov 15 '13 at 12:34

3 Answers
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The dragoons were armed with straight sabers and muskets. Their muskets were longer and had longer range of fire than light cavalry's carbines. While a light cavalryman's eqipment included a carbine sling as a means of keeping his weapon readily available for use, the greater length of musket issued to dragoons made a sling impractical. Thus the stock of the musket was seated in a boot attached to the saddle, and irs barrel restrained by a strap attached to the pommel. When the dragoons expected to go into action they drew sabers and muskets slung on their backs. In 1814 they gave away their long muskets for the infantry.

Napoleon had problems to find the right horses for his dragoons. In 1805 approximately 6.000 of them were without mounts and were organized into 4 foot dragoon regiments. Their duty was to guard the artillery reserves and the baggage trains. After the 1805-campaign Napoleon mounted the foot dragoons on captured Austrian horses. Then after the 1806-campaign Napoleon mounted the rest of the "walkers" on captured Prussian and Saxon horses. The hardships of war in Spain, plus poor horsecare killed thousands of dragoons' mounts. For example in May 1811 the 3e Dragons had only 139 horses left out of 563 ! The situation was so desperate that in 1812 was issued an order that all officers in infantry regiments have to give their horses to the dragoons.

The cavalry of the line was composed of Dragoons [...] in practice they were increasingly used as battle cavalry, though they also acted as flank guards, and when needed, as in Spain, in their original role as mounted infantry.[page 141]

The cavalry was stationed at the wings and its primary mission was to counter the enemy's horse. On occasion, especially if the opposing infantery was not yet deployed, such as at Rossbach (1757) horse was launched againts foot with excellent results. Also if cavalry drove its opponents off the field and was able to rally [...] it could be hurled against the enemy's unprotected flank or rear. These missions were duties of heavy battle cavalry, cuirassiers and dragoons, while light cavalry were used to...[page 15]

In the first phase of Napoleonic Wars they served on the primary theater of war, in Central Europe, charging in numerous battles and skirmishes. In November 1805 the dragoon brigade under Sebastiani took 2,000 prisoners at Pohrlitz.

Colonel Elting writes: "The assignement was sensible, but troopers caught up in the shuffle remembered that veteran dragoons, who hadn't walked farther in years than the distance from their barracks to the nearest bar, ended up in the dismounted units, while their mounts were assigned to raw recruits. The results were rough on everybody: hospitals filled up with spavined veterans, recruits got saddle sores. Also, J.A. Oyon wrote gleefully, matters turned ugly when mounted and dismounted elements of several regiments bivouaced together. The limping veterans crowded over to check on their old horses and found them neglected, sore-backed, and lame.
Blood flowed freely, if only from rookies' noses."

The natural conflict inherent in the dragoon concept was widely recognized even at the time; namely that to be a fully trained infantry formation, the men most absolutely believe that an infantry square cannot be broken by cavalry; and to be a fully trained cavalry formation, they must absolutely believe that only the very best infantry units can resist a cavalry charge. Clearly any one unit can never hold both beliefs.

In Europe, with it's long tradition of sabres drawn cavalry charges on relatively open battlefields, (Napoleonic Era) dragoons were treated and trained as cavalry who occasionally dismounted. Throughout the continent, Dragoons (except Light Dragoons, who were equivalent to Chasseurs) were treated as Medium cavalry, consisting of larger men on larger horses, but unarmoured, and saw themselves as such.

In North America, the concept of the sabres-drawn cavalry charge never really caught on. This can likely be explained by the absence of a knighthood tradition; the prevalence of rifles instead of muskets as dominant firearm; and the much greater cover on the battlefields that increased the importance of, and need for, skirmishers. In consequence, North American Dragoons were generally treated as, and saw themselves as, mounted infantry.

In the days of blade weapons, a cavalryman (post-stirrup) mounted on a horse carrying a sword or spear had greater momentum, and hence greater value than a similarly equipped infantryman without a horse.

In the days of (slow-loading, single shot) muskets, there was a balance. The infantryman had superior firepower, the cavalry man with the blade weapon had the superior speed. To partially compensate for inferior firepower, cavalrymen were given carbines, shorter lighter guns than muskets (or rifles). Such horsemen with both fire- and blade- weapons were known as dragoons. Overall, the combination of horse, sword, and carbinegave a cavalryman a slight advantage over a musketeer, even though none of these weapons used singly, or in pairs, would do so.

The introduction of the rifle, a gun with long range that became a "repeating" weapon around the middle of the 19th century, changed the equation further. Now a cavalryman with a blade weapon was at a clear disadvantage to an infantryman with a rifle. (And rifles couldn't be used properly by horsemen controlling a horse). Cavalry's only advantage was its greater speed; you could get troops to a critical point faster than with infantry. But once "arrived," cavalry fought at a disadvantage, insofar as one out of four men had to "hold the horses" of three others, while those others dismounted and did the actual fighting on foot. So then the equation became, was a cavalry unit with 3/4 strength arriving in a "timely" fashion more valuable than a full-strength infantry unit that arrived later? (Confederate) cavalry commander Nathan Bedford Forrest answered in the affirmative when he said, "I get there fastest with the mostest."